Evolution of the Torsional Rigidity under Geometric Flows

TL;DR

This paper investigates how torsional rigidity of domains changes under various geometric flows, deriving bounds and comparison inequalities for specific manifolds and hypersurfaces.

Contribution

It provides new bounds on torsional rigidity during Ricci and Inverse Mean Curvature Flows for particular geometric settings.

Findings

Bounds on torsional rigidity under Ricci Flow for Heisenberg spaces.

Bounds under Inverse Mean Curvature Flow for convex hypersurfaces.

Comparison inequalities with flat disks for volume and torsional rigidity.

Abstract

This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and homogeneous spheres. Additionally, we establish bounds under the Inverse Mean Curvature Flow for strictly convex, free-boundary, disk-type hypersurfaces within a ball. In this latter case, by extending the analysis to the maximal existence time of the flow, we obtain inequalities of comparison with the flat disk for both volume and torsional rigidity.